Question: Solve the exponential equation for $x$. 81 x + 7 9 5 x − 9 = 9 4 x + 1 \dfrac{81\^{ x+7}}{9\^{ 5x-9}}=9\^{ 4x+1} $x=$
The strategy Let's write $81$ in base $9$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $9$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 81 x + 7 9 5 x − 9 = ( 9 2 ) x + 7 9 5 x − 9 = 9 2 x + 14 9 5 x − 9 = 9 2 x + 14 − ( 5 x − 9 ) = 9 − 3 x + 23 ( 81 = 9 2 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{81\^{ x+7}}{9\^{ 5x-9}}&=\dfrac{(9^2)\^{ x+7}}{9\^{ 5x-9}}&&&&(81=9^2) \\\\\\\\ &=\dfrac{9\^{ C{2x+14}}}{9\^{ {5x-9}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=9\^{ C{2x+14} \ - \ ({5x-9})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=9\^{-3x+23} \end{aligned} Solving the equation We obtain the following equation. 9 − 3 x + 23 = 9 4 x + 1 9\^{ -3x+23}=9\^{ 4x+1} Now we can equate the exponents and solve for $x$. $\begin{aligned} -3x+23&=4x+1\\\\ x &=\dfrac{22}{7}\end{aligned}$ The answer The answer is $x=\dfrac{22}{7}$. You can check this answer by substituting $\it{x=\dfrac{22}{7}}$ in the original equation and evaluating both sides.